The calculus of homotopy functors provides a systematic way to approximate a given functor (say from based spaces to spectra) by so-called `polynomial' functors. Each functor F that preserves weak equivalences has a `Taylor tower' (analogous to the Taylor series of ordinary calculus) which in turn is built from homogeneous pieces that are classified by certain `derivatives' for F. I will review this material and consider the problem of how the Taylor tower of F can be reconstructed from its derivatives. We will discuss some important examples built from mapping spaces. Then. if time permits, I will us this approach to give a classification of analytic functors from based spaces to spectra and try to describe some connections to the Goodwillie-Weiss manifold calculus

: I will give an introduction to Morita theory in stable homotopy theory. This will include the context of differential graded algebra and of spectral algebra. I will also survey some recent related results.

Topological cyclic homology is a topological refinement of Connes' cyclic homology. It was introduced twenty-five years ago by Bökstedt-Hsiang-Madsen who used it to prove the K-theoretic Novikov conjecture for discrete groups all of whose integral homology groups are finitely generated. In this talk, I will give an introduction to topological cyclic homology and explain how results obtained in the intervening years lead to a short proof of this result in which the necessity of the finite generation hypothesis becomes transparent. In the end I will explain how one may hope to remove this restriction and discuss number theoretic consequences that would ensue.

In this talk, I will given an overview of recent work on formulating the structural properties of algebraic K-theory using the framework for studying homotopical categories provided by the development of higher category theory

Models for homotopical categories, known as $(\infty,1)$-categories, are now fairly well-developed. Comparisons are in place between them, and there is a wide variety of applications ranging from topology to algebra to algebraic geometry. Generalizing them to models for homotopical higher categories, or $(\infty,n)$-categories, requires substantially greater technicality. Much work has been done in this direction, due to recent work of many authors, but the full picture is still very much work in progress. In this talk we will review different models for $(\infty,1)$-categories and discuss the number of ways they are being generalized to models for $(\infty,n)$-categories, as well as the known and conjectured comparisons between them

We will begin with an introduction to stable motivic homotopy theory. Then we will use the motivic version of the Adams spectral sequence to compute stable motivic homotopy groups. We will discuss a number of open questions concerning these computations. Along the way, we will encounter some new results about classical stable homotopy groups and equivariant stable homotopy groups.

This will be a survey talk on subjects related to the J-homomorphism. In addition to the original geometric construction which assigns to a vector bundle its associated spherical fibration, we will touch on more algebraic constructions derived form it, in particular via Picard and unit spectra. We will examine the map from a computational point of view, studying its p-localisation and interpretation in terms of K(1)-local homotopy theory. We will almost certainly fail to do justice to any of these ideas in the time available. At the end, we will briefly extend some of these results to a higher chromatic setting.

Local structure of groups and of their classifying spaces
Bob Oliver (Université de Paris XIII (Paris-Nord))

Location

MSRI: Simons Auditorium

Video

Abstract

This will be a survey talk on the close relationship between the local structure of a nite group or compact Lie group and that of its classifying space. By the p-local structure of a group G, for a prime p, is meant the structure of a Sylow p-subgroup S G (a maximal p-toral subgroup if G is compact Lie), together with all G-conjugacy relations between elements and subgroups of S. By the p-local structure of the classifying space BG is meant the structure (homotopy properties) of its p-completion BG^p . For example, by a conjecture of Martino and Priddy, now a theorem, two nite groups G and H have equivalent p-local structures if and only if BG^p ' BH^p . This was used, in joint work with Broto and Møller, to prove a general theorem about local equivalences between nite Lie groups a result for which no purely algebraic proof is known. As another example, these ideas have allowed us to extend the family of p-completed classifying spaces of (nite or compact Lie) groups to a much larger family of spaces which have many of the same very nice homotopy theoretic properties.

The homotopy theory of compact Lie groups is very well understood by now. The rich structure of these groups (for example: existence and uniqueness of maximal tori, corresponding Weyl groups etc.) may be exploited to classify these groups. This classification even extends to homotopical versions of these groups known as p-compact groups. In the last few decades a beautiful new class of (non-compact) topological groups has been constructed. These are known as Kac-Moody groups and they share most of the structure that compact Lie groups admit. Kac-Moody groups have been shown to be relevant in mathematical physics and further investigation by several mathematicians (including the speaker) seems to suggest that Kac-Moody groups are surprisingly amenable to homotopical techniques. This makes these groups prime candidates for study from the standpoint of homotopy theory

: Topological modular forms and its generalizations are objects in stable homotopy that realize a connection to 1-dimensional formal group laws. In this talk I'll describe how this perspective on stable homotopy has emerged, and how using algebraic geometry is providing us with a library of new objects in homotopy theory

I will give an introduction to the theory of representation stability, through the lens of its applications in homological stability. I'll focus on three applications: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; and stability for twisted homology such as H_i( GL_n(R); R^n ), where the coefficients change along with the groups.

I will first discuss some fairly classical homological stability phenomena: spaces of 0-manifolds (e.g. configuration spaces), and spaces of 2-manifolds (e.g. Riemann's moduli space). I will explain the general method, introduced by Quillen, for proving these stability theorems, and then explain some recent work with Søren Galatius which proves such a stability theorem for moduli spaces of 2n-dimensional manifolds (for n > 2).

I will discuss recent joint work with Oscar Randal-Williams, aimed at calculating the cohomology of BDiff(W) and related spaces, where W is a smooth 2n-dimensional manifold, Diff(W) is the topological group of diffeomorphisms of W, and BDiff(W) is its classifying space. Surprisingly, the cohomology ring turns out to be partially independent of W through a range of degrees (homological stability). In this talk, I will discuss how infinite loop spaces can be used to describe the cohomology in this stable range.